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As shown by Morse and Feshbach (1953), the Helmholtz Differential Equation is separable in Confocal Paraboloidal Coordinates.
See also Confocal Paraboloidal Coordinates
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, p. 664, 1953.